Use Cramer's Rule to solve each system.
$\{\begin{array}{c}7x+2y=0\\ 2x+y=-3\end{array}$

Verified step by step guidance

1. Identify the system of linear equations you need to solve. Typically, it will be in the form: \[\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\]
2. Write down the coefficient matrix \[A\], which consists of the coefficients of the variables: \[A = \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}\]
3. Calculate the determinant of the coefficient matrix \[A\], denoted as \[\det(A)\], using the formula: \[\det(A) = a_1b_2 - a_2b_1\]
4. Form the matrices \[A_x\] and \[A_y\] by replacing the respective columns of \[A\] with the constants from the right side of the equations: \[A_x = \begin{bmatrix} c_1 & b_1 \\ c_2 & b_2 \end{bmatrix}, \quad A_y = \begin{bmatrix} a_1 & c_1 \\ a_2 & c_2 \end{bmatrix}\]
5. Calculate the determinants \[\det(A_x)\] and \[\det(A_y)\], then use Cramer's Rule to find the solutions for \[x\] and \[y\]: \[x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}\]

Key Concepts

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies to square systems where the number of equations equals the number of variables. Each variable is found by replacing the corresponding column of the coefficient matrix with the constants vector and dividing the determinant of this new matrix by the determinant of the coefficient matrix.

Determinants

A determinant is a scalar value computed from a square matrix that provides important properties about the matrix, such as invertibility. For Cramer's Rule, the determinant of the coefficient matrix must be nonzero to ensure a unique solution exists. Determinants can be calculated using expansion by minors or row operations.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same set of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Understanding how to represent these systems in matrix form is essential for applying methods like Cramer's Rule.